The moduli space of holomorphic normal projective connections is an affine space, as it is identified with the collection of all holomorphic 1-cocycles whose coboundary is a suitable ``traceless Atiyah class'' of the tangent bundle. The paper 

Robert Molzon and Karen Pinney Mortensen, 
*The Schwarzian derivative for maps between manifolds with complex projective connections*, **Trans. Amer. Math. Soc.** 348 (1996), no. 8, 3015–3036. MR 1348154 (96j:32028) 27, 55

is perhaps the best introduction to the theory of complex projective connections. I don't know a good reference for the relation to the Atiyah class, though it appears somewhere in the work of Kobayashi. The moduli space of flat holomorphic projective connections on a compact complex manifold is a complex subvariety, not known to be smooth.